Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=-1, a_{2}=4 $$

Step-by-Step Solution

Answer

The first five terms are \( -1, 4, -16, 64, -256 \).

Step 1: Understand a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted as \( r \).

Step 2: Find the Common Ratio

The common ratio \( r \) can be found by dividing the second term \( a_2 \) by the first term \( a_1 \). Substituting the given values, \( r = \frac{a_2}{a_1} = \frac{4}{-1} = -4 \).

Step 3: Calculate the Third Term

The third term \( a_3 \) of the sequence is obtained by multiplying the second term \( a_2 \) by the common ratio \( r \). That is, \( a_3 = a_2 \times r = 4 \times -4 = -16 \).

Step 4: Calculate the Fourth Term

The fourth term \( a_4 \) is calculated by multiplying the third term \( a_3 \) by the common ratio \( r \). Therefore, \( a_4 = a_3 \times r = -16 \times -4 = 64 \).

Step 5: Calculate the Fifth Term

The fifth term \( a_5 \) is found by multiplying the fourth term \( a_4 \) by the common ratio \( r \). Consequently, \( a_5 = a_4 \times r = 64 \times -4 = -256 \).

Step 6: Write the First Five Terms

Having calculated all necessary terms, the first five terms of the sequence are \( -1, 4, -16, 64, -256 \).

Key Concepts

Common Ratio

In a geometric sequence, the concept of a common ratio is fundamental. This is the fixed number that we multiply by to get from one term in the sequence to the next. If you know the first two terms of the sequence, like in this example with the first term (\(a_1 = -1\)) and the second term (\(a_2 = 4\)), you can find the common ratio.
To find the common ratio, divide the second term by the first term. Here, it looks like this:

• Common Ratio (\(r\)) = \(\frac{a_2}{a_1} = \frac{4}{-1} = -4\)

This negative common ratio indicates that the sequence alternates signs as we progress through it. Each term is four times the previous term, but with the opposite sign.
This simple division helps you unlock the mystery of how each term relates in the series, making it the cornerstone of understanding geometric sequences.

Sequence Terms

Understanding sequence terms in a geometric sequence revolves around how each term is generated using the common ratio. Once we have the common ratio, calculating subsequent terms is straightforward. Let's look at how this works in practice, using the sequence from the problem:
We already know:

• First term (\(a_1\)) = -1
• Second term (\(a_2\)) = 4

To find the next terms:

• Third term (\(a_3\)): Multiply the second term by the common ratio:
  \(a_3 = a_2 \times r = 4 \times -4 = -16\)
• Fourth term (\(a_4\)): Multiply the third term by the common ratio:
  \(a_4 = a_3 \times r = -16 \times -4 = 64\)
• Fifth term (\(a_5\)): Multiply the fourth term by the common ratio:
  \(a_5 = a_4 \times r = 64 \times -4 = -256\)

This process of using the common ratio to find new terms can be repeated infinitely, showing how powerful this simple procedure is in finding any term of a geometric sequence.

Multiplying Series

The concept of multiplying series in a geometric sequence is both elegant and practical. Since each term is a multiple of the previous one by a fixed amount, or common ratio, identifying these terms enables us to easily build a series.
This method applies to any geometric sequence consistently:

• Identify or calculate the common ratio.
• Compute each subsequent term by multiplying the previous term by the common ratio.

For example, in our sequence:

• Starting from the first term \(a_1 = -1\), multiply by the common ratio \(-4\) to get the second term \(a_2 = 4\).
• Repeat this multiplication to progress through the sequence. Each multiplication turns the term into the next sequence term: \(-1 \rightarrow 4 \rightarrow -16 \rightarrow 64 \rightarrow -256\).

This technique ensures we can continue this process to find the nth term. It demonstrates the simplicity behind geometric sequences and shows how once you understand the rule of multiplying by the common ratio, infinite terms can be generated efficiently.